Introduction to Time Series Analysis 113
following period’s stock price conditional on this period’s information will
contain a systematic error. The model to be tested is, then,
SFtt+ 10 =+αα 1 ()t+εwith a potential nonzero linear trend captured by α 0. A fair price would be if
the estimates are a 0 = 0 and a 1 = 1. Then, the markets would be in approxi-
mate equilibrium. If not, the forward prices have to be adjusted accordingly
to prohibit predictable gains from the differences in prices.
The methodology to do so is the so-called error correction model in the
sense that today’s (i.e., this period’s) deviations from the equilibrium price
have to be incorporated into tomorrow’s (i.e., the following period’s) price
to return to some long-term equilibrium. The model is given by the equa-
tions system
SS SFt
FtFttt++=−()tt++− +>
+=21 120
1
αε() , α
()())(+−βξ()SFtt++ 11 t),+>β 0with
Et
Ett
t[|]
[|]
ε
ξ+
++=
=
2
110
0
At time t + 2, the term α()SFt+ 1 − ()t in the price of St+ 2 corrects for devia-
tions from the equilibrium ()SFt+ 1 − ()t stemming from time t + 1. Also, we
adjust our forward price F(t + 1) by the same deviation scaled by β. Note that,
now, the forward price, too, is affected by some innovation, ξt+ 1 , unknown
at time t. In contrast to some disturbance or error term ε, which simply rep-
resents some deviation from an exact functional relationship, the concept
of innovation such as in connection with the ξt+ 1 is that of an independent
quantity with a meaning such as, for example, new information or shock.
In general, the random walk and error correction models can be esti-
mated using least squares regression introduced in Chapter 2. However, this
is only legitimate if the regressors (i.e., independent variables) and distur-
bances are uncorrelated.
Key Points
■ (^) A sequence of observations which are ordered in time is called a time series.
■ (^) Time series are significant in modeling price, return, or interest rate pro-
cesses as well as the dynamics of economic quantities.